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Let $L$ be a Lie algebra of dimension $n$ and let $M(n \times n)$ denote the space of square matrices of size $n$. We know that the adjoint representation is a homomorphism of Lie algebras $\text{ad} : L \to M(n \times n)$. How many more representations are there? What is the dimension of $\text{Hom} \big( L, M(n \times n) \big)$?

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Concerning the first question, there are in general many more representations than just the adjoint representation. The adjoint representation is special. For once, we have $\mathrm{ad}(L)\subseteq \mathrm{Der}(L)$, which in general is only a subspace of $\mathfrak{gl}_n(K)$, the Lie algebra of $M_n(K)$. Secondly, for a nilpotent Lie algebra, the adjoint representation is never faithful. So taking any faithful representation of $L$ of the same dimension as the adjoint representation (if it exists) already gives a different representation. Finally, the example of the $3$-dimensional Heisenberg Lie algebra shows that we have many $3$-dimensional representations in this case.

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